From d0ec6464cce9a5f6d417d675741aa546ddcb6144 Mon Sep 17 00:00:00 2001 From: DylanVsn <43576618+DylanVsn@users.noreply.github.com> Date: Fri, 7 Feb 2020 15:45:32 +0100 Subject: [PATCH] =?UTF-8?q?=C3=A0=20jour?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 3/TP3_voisin.py | 13 +++++- 4/tp4.pdf | Bin 0 -> 141677 bytes 4/tp4_Voisin.py | 114 ++++++++++++++++++++++++++++++++++++++++++++++++ 3 files changed, 126 insertions(+), 1 deletion(-) create mode 100644 4/tp4.pdf create mode 100644 4/tp4_Voisin.py diff --git a/3/TP3_voisin.py b/3/TP3_voisin.py index 879c694..9ba9763 100755 --- a/3/TP3_voisin.py +++ b/3/TP3_voisin.py @@ -253,4 +253,15 @@ def main(argv): if __name__ == "__main__": - main(sys.argv[1:]) + # main(sys.argv[1:]) + m = np.array( + [ + [2, 3], + [3, 2] + ] + ) + u1, u2, l1, l2 = diagonaliser_matrice22(m) + u1 /= np.linalg.norm(u1) + u2 /= np.linalg.norm(u2) + print(u1, u2) + print(l1, l2) \ No newline at end of file diff --git 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zn-|&tJM1f_V9Th%h;cdz@2cS##nY(I+LtbYaU#Wb(oM1N-zrt;@pjUO{q(t8v`|NAAnK|rn$uU7zu{z{u(i4r;kgN&*y`u_l4%i-4m literal 0 HcmV?d00001 diff --git a/4/tp4_Voisin.py b/4/tp4_Voisin.py new file mode 100644 index 0000000..e0014de --- /dev/null +++ b/4/tp4_Voisin.py @@ -0,0 +1,114 @@ +#!/usr/bin/env python3 + +""" + auteur: Dylan Voisin - M1 Informatique Luminy + MNI: TP 4 +""" + +import numpy as np +from numpy.polynomial import polynomial as P +from copy import deepcopy as dp +from math import isclose + +def vect_propre(A, µ=1, nb_iter=15): + I = np.identity(A.shape[0]) + v = np.ones(A.shape[0]) + inv = np.linalg.inv(A - µ*I) + for i in range(nb_iter): + v = inv @ v / np.linalg.norm(inv@v) + return v + +def vect_propre2(A, eps = 0.1): + v = np.random.random(A.shape[0]) + v /= np.linalg.norm(v) + lambda_ = v.transpose() @ A @ v + eps = 1e-13 + écart = float('inf') + while écart > eps: + w = lambda_ * A @ v + v = w / np.linalg.norm(w) + old = lambda_ + lambda_ = (v.transpose() @ A @ v) / (v.transpose() @ v) + écart = abs(old - lambda_) + return v + +def vect_propre_généralisé(A, eps = 1-9): + A_ = A @ A.transpose() + v = vect_propre2(A_, eps) + v = (v.transpose() @ A @ v) / (v.transpose() * v) + return v + +def eval_poly(p, x): + """un polynome sera représenté par une liste pleine de ses coefficients + f = a_0 + a_1 * x ** 1 + … + a_n * x ** b_n + [a_0, a_1, a_2, …, a_n] + """ + return sum(p[i] * x ** i for i in range(len(p))) + +def dichotomie(size, step): + l = [0, size] + lc = dp(l) + for i in range(step): + for j in range(len(l)-1): + m = (l[j+1] + l[j]) / 2 + lc.insert(2*(j+1)-1, m) + yield m + l = dp(lc) + +def newton(p, epsilon_f = 1e-13, epsilon_c = 1e-9): + """peut faire une boucle infinie à cause de l'imprécision + à régler avec les paramètres epsilon, mais l'imprécision va + se répercuter sur les valeurs des racines""" + x = np.random.random() * 100 + pp = np.polyder(p[::-1])[::-1] + roots = [] + while True: + # toujours de l'imprécision à gérer + if abs(eval_poly(p, x)) < epsilon_f: + roots.append(x) + p = P.polydiv(p, (-x, 1))[0] + # on purge l'imprécision + p = list(map(lambda x: x if abs(x) > epsilon_c else 0, p)) + pp = np.polyder(p[::-1])[::-1] + if len(p) == 1: + return roots # f = c + fpx = eval_poly(pp, x) + if fpx == 0: + print(x) + # extremum (local ou non) + old = x + step = 5 + maxrange = 1000 + eps = 1*10**(-8) + sg = np.sign(eval_poly(pp, x-eps)) + sd = np.sign(eval_poly(pp, x+eps)) + for diff in dichotomie(maxrange, step): + if np.sign(eval_poly(pp, x-diff)) != sg: + x = x - diff + break + elif np.sign(eval_poly(pp, x+diff)) != sd: + x = x + diff + break + if x == old: + return roots # on considère qu'on a toutes les racines + x -= eval_poly(p, x) / eval_poly(pp, x) + return roots + +if __name__ == '__main__': + m = np.array( + [ + [2, 3], + [3, 2] + ] + ) + v = vect_propre(m) + v /= np.linalg.norm(v) + print(v) + v = vect_propre2(m) + print(v) + v = vect_propre_généralisé(m) + v /= np.linalg.norm(v) + print(v) + # p = (0, 1, -2) # -2x²+x + p = (30, -19, -15, 3, 1) # (x+5)(x-3)(x-1)(x+2) + print(newton(p)) \ No newline at end of file